Wasserstein Distance Calculator (Earth Mover's Distance)

The Wasserstein-1 distance measures the cost of the optimal transport plan that turns one distribution into another — the minimum of (mass moved) × (distance moved), summed over every unit of probability. It comes from the Kantorovich–Rubinstein formulation of optimal transport, and in the vision literature the same quantity is Rubner, Tomasi & Guibas's Earth Mover's Distance. In one dimension the general linear program collapses to an exact closed form based on the cumulative distribution functions (CDFs):

W₁(P, Q) = ∫ |F_P(x) − F_Q(x)| dx

For a shared, ordered support x₁ < x₂ < … < xₙ with normalised masses, the integral becomes a finite sum over the gaps between support points:

1. Normalise each vector so it sums to 1 (Wasserstein-1 needs equal total mass). This lets you paste raw counts as well as probabilities.
2. Build the cumulative masses P(i) = Σ₍ⱼ≤ᵢ₎ p̂ⱼ and Q(i) = Σ₍ⱼ≤ᵢ₎ q̂ⱼ at each support point.
3. Sum |P(i) − Q(i)| · (x₍ᵢ₊₁₎ − xᵢ) over the intervals. With unit-spaced bins (Δx = 1) this reduces to Σ |P(i) − Q(i)|.
4. For raw samples, give every value mass 1/n and apply the same integral over the pooled sorted support. When the two samples are equal length it equals the order-statistic mean, (1/n) Σ |A₍ᵢ₎ − B₍ᵢ₎|.

The tool verifies the answer a second, independent way. The same distance equals the integral of the gap between the inverse CDFs (the quantile functions), W₁ = ∫₀¹ |F_P⁻¹(u) − F_Q⁻¹(u)| du— the comonotonic transport plan. The two integrals are equal by Fubini's theorem, so the calculator computes both and shows them reconciling. A third built-in signal is the bound W₁ ≥ |E[P] − E[Q]|: the mean can never move further than the total transport. Unlike KL divergence, W₁ is a true metric — symmetric and satisfying the triangle inequality — and it stays finite even when the two supports do not overlap, which is exactly why it is the drift metric of choice in model monitoring and Wasserstein GANs.

• Wasserstein metric — definition and the 1-D CDF closed form (Kantorovich–Rubinstein)
• Rubner, Tomasi & Guibas (1998), “A Metric for Distributions…” (Earth Mover's Distance)
• Arjovsky, Chintala & Bottou (2017), “Wasserstein GAN”, arXiv:1701.07875
• SciPy — scipy.stats.wasserstein_distance (reference 1-D implementation)

The formula and worked examples on this page were last reconciled against SciPy on 2026-07-16. The closed form is a mathematical identity, not a published rate, so it does not change over time — the date records when the numbers were re-checked.